# Lopatin A. K.

### Normalization and averaging on compact lie groups in nonlinear mechanics

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 47–67

We consider the method of normal forms, the Bogolyubov averaging method, and the method of asymptotic decomposition proposed by Yu. A. Mitropol’skii and the author of this paper. Under certain assumptions about group-theoretic properties of a system of zero approximation, the results obtained by the method of asymptotic decomposition coincide with the results obtained by the method of normal forms or the Bogolyubov averaging method. We develop a new algorithm of asymptotic decomposition by a part of the variables and its partial case — the algorithm of averaging on a compact Lie group. For the first time, it became possible to consider asymptotic expansions of solutions of differential equations on noncommutative compact groups.

### Bogolyubov averaging and normalization procedures in nonlinear mechanics. IV

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1044–1068

In this paper, we apply the theory developed in parts I-III [*Ukr. Math. Zh.*,**46**, No. 9, 1171–1188; No. 11, 1509–1526; No. 12, 1627–1646 (1994)] to some classes of problems. We consider linear systems in zero approximation and investigate the problem of invariance of integral manifolds under perturbations. Unlike nonlinear systems, linear ones have centralized systems, which are always decomposable. Moreover, restrictions connected with the impossibility of diagonalization of the coefficient matrix in zero approximation are removed. In conclusion, we apply the method of local asymptotic decomposition to some mechanical problems.

### Bogolyubov averaging and normalization procedures in nonlinear mechanics. I

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1171–1188

We suggest a new method for asymptotic analysis of nonlinear dynamical systems based on group-the-oretic methods. On the basis of the Bogolyubov averaging method, we develop a new normalization procedure — “asymptotic decomposition.” We clarify the contribution of this procedure to the interpretation and development of the averaging method for systems in the standard form and systems with several fast variables. According to this method, the centralized system is regarded as a direct analog of the system averaged in Bogolyubov's sense. The operation of averaging is interpreted as the Bogolyubov projector, i.e., the operation of projection of an operator onto the algebra of centralizer.

### Asymptotic decomposition of completely integrable pfaffian systems with small parameter

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1988. - 40, № 3. - pp. 349-356

### Basis of the algorithm of asymptotic decomposition for a finite number of approximations

Ukr. Mat. Zh. - 1987. - 39, № 6. - pp. 732-737

### Asymptotic decomposition of differential systems with a small parameter

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1987. - 39, № 2. - pp. 194-204

### Asymptotic decomposition of differential systems with a small parameter in the representation space of a finite-dimensional Lie group

Ukr. Mat. Zh. - 1987. - 39, № 1. - pp. 56–64

### Asymptotic decomposition of systems of nonlinear ordinary differential equations

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1984. - 36, № 1. - pp. 35 - 45